4.4 Numerical Examples
4.4.2 Implementation for GMAB
Figure 4.2 illustrates the optimal hedging strategy for the GMAB liability put option in (4.1) and the distributions of the market index under the Heston-type stochastic volatility, given the market parameters: S0 =F0 =GT = 100, ρ=−0.3,κ∗ = 3, ¯v∗ = 0.02, µ= 0.15, r = 0.05, σ = 0.5, λ=−0.15, T = 1, αL= 0.05, αH = 0.95, M = 50,000, Ks =Kz = 150 with the selected pair of a = 0.1513 and b = 1.5. The constant c determined in Step (S4) is 0.225. The numerical results show that the optimal hedging optionhopt outperforms its counterpart, the mean-square one hms. Forµ= 0.15, the expected shortfalls fromhopt and hms are 0.6746 and 0.9663 respectively, which is a reduction of 30.2% in shortfall risk. The
standard deviation of the shortfall risk fromhopt is reduced by 23.72% when compared to hms. 70 80 90 100 110 120 130 140 150 0 10 20 30 40 50 60 hL hU hopt − option h ms − option 0 50 100 150 200 250 0 0.005 0.01 0.015 0.02 0.025 density under P density under Q
Figure 4.2: The optimal static hedging option for the GMAB liability put option (GT −FTs)
+
(red line) conditional on ST = s in the left panel and the probability den- sities of the underlying ST with the Heston-type stochastic volatility in the right panel.
4.5
Concluding Remarks
The main objective of this chapter is to extend a theoretical result that characterizes an optimal hedging strategy for a path-dependent GMAB liability in the incomplete Heston market. The presented numerical results demonstrate both the feasibility of the optimal hedging method. The optimal static hedging strategy can be considered as a benchmark, when compared to other strategies with the introduction of the stochastic volatility and the path-dependent feature on the payoff. We take the GMAB liability put option as an example but the proposed hedging strategy is not limited to that. For example, the construction of simplified hedges and most VA liabilities discussed in this thesis can result in the hedging problem for the path-dependent contracts with the stochastic volatility. For the reverse mortgage contracts, the proposed optimal hedging options can be developed for the complexity of the crossover risk, such as hedging with the conditioning event(s) under the environment of either the stochastic volatility or the stochastic interest rate.
Chapter 5
Conclusions and Future Research
In this thesis, we explored pricing and hedging issues on various innovative products in finance and insurance. The innovations of pricing fulfilled the rising market demands for the products with path-dependent features, such as simplified derivatives, variable annuities and reverse mortgage contracts. To hedge these products, we relied on the method by
Kolkiewicz(2016) and extended the results that can be applied in different angles. In this chapter, we propose several lines of future research.
The first direction lies in developing an optimal static hedging strategy for the simplified derivatives. In Chapter 1, we extended the work of Bernard, Moraux, R¨uschendorf, and Vanduffel(2015) to construct a simplified alternative that resembles a given highly path- dependent derivative by adding multiple conditional benchmarks. A natural extension is to construct the simplified alternatives under different criteria in the context of minimizing its expected shortfall and the Value-at-Risk to the original derivative. Due to the nature of the simplified derivative, we can generalize the corresponding construction by conditioning on more benchmark states for improving the performance to meet the given criteria. Since the simplified derivative is constructed by only preserving certain distributional properties of the original payoff, the proposed construction can be expected to provide a better hedging performance to the original payoff than the simplified hedge only keeping its distributional property. For hedging the simplified derivative itself, we will compare the hedging performance of the optimal static hedging strategy to the simplified alternative with that of the semi-static hedging as proposed in Chapter1, which only involves a limited number of basis options. The optimal static hedging option for the simplified derivative can be constructed either in the Black-Scholes framework or more general markets with stochastic volatility.
The second area of future research might consider the effect of state-dependent fees or the state-dependent structure on more complex path-dependent VA products, such as guaranteed minimum withdrawal benefits (GMWBs) and flexible premium in variable annuities (FPVAs) introduced byBernard, Cui, and Vanduffel(2017). Both these contracts allow the policyholder to make withdrawals or additional payment contributions to the underlying account of the variable annuity depending on the path of the fund value. In Chapter 2, the benefits of a GMAB contract tied with the volatility-dependent fees have been examined. In general, the affine models and dynamic programming framework offer the possibility to involve more path-dependent features such as the state-dependent fees tied with the market volatility, underlying processes or surrender strategies in most recent VA contracts. Charging a state-dependent fee cannot exclude the possibility of the lapse of the contract, but we expect that it will make the optimal surrender boundary lower than that of charging a fixed fee rate. In the study of GMWBs, we can consider the strategy of optimal withdrawals in the presence of state-dependent fees. In the study of FPVAs, we will study the optimal flexible premiums depending on market conditions such as the underlying fund and the interest rates, while most current academic research assume that there is a single premium payment independent of market conditions. We expect that such flexible premiums can be optimized for covering the costs of liabilities. By modifying the state-dependent premiums to the fund return in FPVAs, this problem is comparable to the problem of optimal withdrawals by the policyholder in a GMWB where the withdrawal rate is subtracted from the fund return. Adding the state-dependency to the VA contracts leads to the complexity of the hedging strategy. Thus, the corresponding optimal hedging strategy will be developed for accommodating the hedging needs.
The third direction is to design the defaultable reverse mortgage (RM) contracts in more general markets and to study the application of the optimal static hedging on the reverse mortgage markets. Our goal is not limited to hedging a particular defaultable contract: more general credit-risk derivatives can be considered. In Chapter3, a novel pricing of the RM contract was proposed for improving the solvency of the current HECM program in the presence of the borrower’s default risk. Despite our pricing scheme designed for reducing the lender’s hedging difficulty, it cannot replace the existing hedging program. To hedge a defaultable RM contract, we require an analysis of optimal hedging in a setting where the payoff of the crossover risk and the corresponding conditioning event are dependent and have to be priced and hedged accordingly. Since the methods by Kolkiewicz (2016) allow the generosity of dependency, it is possible to build the optimal static hedging option on the conditioning events, at which the crossover risk arises from the events of default and death. Existing literature such as Bertus, Hollans, and Swidler (2008) and Fabozzi, Shiller, and Tunaru (2010) has shown that the house price risk is difficult to hedge due
to the basis risk between house price index and the corresponding future contracts offered by the Chicago Mercantile Exchange (CME). The optimal hedging strategy, allowing the inclusion of basis risk, is then motivated for a solution to minimize the expected shortfall for the crossover risk using the CME housing future contracts. In Chapter3, we formulated the RM pricing problem under the Black-Scholes framework but it can also be extended to more complex settings by involving stochastic volatility and stochastic interest rates.
For problems in Chapters 2 - 4, we will consider other models, like those based on L´evy processes. Similarly, hedging problems can also be considered for L´evy processes. For example, Alonso-Garc´ıa, Wood, and Ziveyi (2018) extend the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with GMWB riders. They demonstrate superior computational efficiency of the COS method to price and hedge the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of L´evy processes. The framework developed is general enough to incorporate complex policyholder behavior decisions and sophisticated contract features such as the reset provision. They further use the framework to investigate the risk minimisation hedg- ing strategies under the concepts outlined inKolkiewicz and Liu (2012). Our problems in Chapters2-4can be further generalized in at least two ways. The first one is to search for the optimal solutions in both the stage of pricing, such as allowing the static and dynamic policyholder withdrawal behaviour in GMWBs, and the stage of building the optimal static hedging strategies in the spirit ofKolkiewicz (2016), for more complex path-dependent VA products. To tackle the complexity, we will adopt a backward recursive dynamic program- ming algorithm in aid of the COS method as proposed byAlonso-Garc´ıa, Wood, and Ziveyi
(2018). The second generalization is to model the defaultable reverse mortgage contracts in the L´evy markets and to price and hedge accordingly with the conditioning events under more realistic credit-risk modeling. In Chapter3, we have developed a valuation model for determining the annuity loan payment based on the levels of default risk, assuming that the default probability is modeled with an average hazard rate. Further, more realistic assumptions in Mei (2016) can be considered for modeling the default risk by assuming that the hazard rate can be stochastically tied with the market conditions such as loan status, interest rate and house price. And then, the annuity loan payment rate can be determined accordingly based on the market conditions. Since the RM contracts allow the lapse by default, it arises the optimal determination for the annuity loan payment in accordance with the default risk based on market conditions, which is in the same spirit of the optimal withdrawals for GMWBs. Pricing and hedging such defaultable RM contracts can be extended in the framework of dynamic programing with the aid of the COS method byAlonso-Garc´ıa, Wood, and Ziveyi (2018).
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